Advanced numerical and semi-analytical methods for differential equations

Examines numerical and semi-analytical methods for differential equations that can be used for solving practical ODEs and PDEs This student-friendly book deals with various approaches for solving differential equations numerically or semi-analytically depending on the type of equations and offers si...

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Detalles Bibliográficos
Otros Autores:	Chakraverty, Snehashish, author (author)
Formato:	Libro electrónico
Idioma:	Inglés
Publicado:	Hoboken, New Jersey : Wiley 2019.
Edición:	1st edition
Materias:	Differential equations.
Ver en Biblioteca Universitat Ramon Llull:	https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009630163006719

Tabla de Contenidos:

Cover
Title Page
Copyright
Contents
Acknowledgments
Preface
Chapter 1 Basic Numerical Methods
1.1 Introduction
1.2 Ordinary Differential Equation
1.3 Euler Method
1.4 Improved Euler Method
1.5 Runge-Kutta Methods
1.5.1 Midpoint Method
1.5.2 Runge-Kutta Fourth Order
1.6 Multistep Methods
1.6.1 Adams-Bashforth Method
1.6.2 Adams-Moulton Method
1.7 Higher‐Order ODE
References
Chapter 2 Integral Transforms
2.1 Introduction
2.2 Laplace Transform
2.2.1 Solution of Differential Equations Using Laplace Transforms
2.3 Fourier Transform
2.3.1 Solution of Partial Differential Equations Using Fourier Transforms
References
Chapter 3 Weighted Residual Methods
3.1 Introduction
3.2 Collocation Method
3.3 Subdomain Method
3.4 Least‐square Method
3.5 Galerkin Method
3.6 Comparison of WRMs
References
Chapter 4 Boundary Characteristics Orthogonal Polynomials
4.1 Introduction
4.2 Gram-Schmidt Orthogonalization Process
4.3 Generation of BCOPs
4.4 Galerkin's Method with BCOPs
4.5 Rayleigh-Ritz Method with BCOPs
References
Chapter 5 Finite Difference Method
5.1 Introduction
5.2 Finite Difference Schemes
5.2.1 Finite Difference Schemes for Ordinary Differential Equations
5.2.1.1 Forward Difference Scheme
5.2.1.2 Backward Difference Scheme
5.2.1.3 Central Difference Scheme
5.2.2 Finite Difference Schemes for Partial Differential Equations
5.3 Explicit and Implicit Finite Difference Schemes
5.3.1 Explicit Finite Difference Method
5.3.2 Implicit Finite Difference Method
References
Chapter 6 Finite Element Method
6.1 Introduction
6.2 Finite Element Procedure
6.3 Galerkin Finite Element Method
6.3.1 Ordinary Differential Equation
6.3.2 Partial Differential Equation
6.4 Structural Analysis Using FEM.
6.4.1 Static Analysis
6.4.2 Dynamic Analysis
References
Chapter 7 Finite Volume Method
7.1 Introduction
7.2 Discretization Techniques of FVM
7.3 General Form of Finite Volume Method
7.3.1 Solution Process Algorithm
7.4 One‐Dimensional Convection-Diffusion Problem
7.4.1 Grid Generation
7.4.2 Solution Procedure of Convection-Diffusion Problem
References
Chapter 8 Boundary Element Method
8.1 Introduction
8.2 Boundary Representation and Background Theory of BEM
8.2.1 Linear Differential Operator
8.2.2 The Fundamental Solution
8.2.2.1 Heaviside Function
8.2.2.2 Dirac Delta Function
8.2.2.3 Finding the Fundamental Solution
8.2.3 Green's Function
8.2.3.1 Green's Integral Formula
8.3 Derivation of the Boundary Element Method
8.3.1 BEM Algorithm
References
Chapter 9 Akbari-Ganji's Method
9.1 Introduction
9.2 Nonlinear Ordinary Differential Equations
9.2.1 Preliminaries
9.2.2 AGM Approach
9.3 Numerical Examples
9.3.1 Unforced Nonlinear Differential Equations
9.3.2 Forced Nonlinear Differential Equation
References
Chapter 10 Exp‐Function Method
10.1 Introduction
10.2 Basics of Exp‐Function Method
10.3 Numerical Examples
References
Chapter 11 Adomian Decomposition Method
11.1 Introduction
11.2 ADM for ODEs
11.3 Solving System of ODEs by ADM
11.4 ADM for Solving Partial Differential Equations
11.5 ADM for System of PDEs
References
Chapter 12 Homotopy Perturbation Method
12.1 Introduction
12.2 Basic Idea of HPM
12.3 Numerical Examples
References
Chapter 13 Variational Iteration Method
13.1 Introduction
13.2 VIM Procedure
13.3 Numerical Examples
References
Chapter 14 Homotopy Analysis Method
14.1 Introduction
14.2 HAM Procedure
14.3 Numerical Examples
References.
Chapter 15 Differential Quadrature Method
15.1 Introduction
15.2 DQM Procedure
15.3 Numerical Examples
References
Chapter 16 Wavelet Method
16.1 Introduction
16.2 Haar Wavelet
16.3 Wavelet-Collocation Method
References
Chapter 17 Hybrid Methods
17.1 Introduction
17.2 Homotopy Perturbation Transform Method
17.3 Laplace Adomian Decomposition Method
References
Chapter 18 Preliminaries of Fractal Differential Equations
18.1 Introduction to Fractal
18.1.1 Triadic Koch Curve
18.1.2 Sierpinski Gasket
18.2 Fractal Differential Equations
18.2.1 Heat Equation
18.2.2 Wave Equation
References
Chapter 19 Differential Equations with Interval Uncertainty
19.1 Introduction
19.2 Interval Differential Equations
19.2.1 Interval Arithmetic
19.3 Generalized Hukuhara Differentiability of IDEs
19.3.1 Modeling IDEs by Hukuhara Differentiability
19.3.1.1 Solving by Integral Form
19.3.1.2 Solving by Differential Form
19.4 Analytical Methods for IDEs
19.4.1 General form of nth‐order IDEs
19.4.2 Method Based on Addition and Subtraction of Intervals
References
Chapter 20 Differential Equations with Fuzzy Uncertainty
20.1 Introduction
20.2 Solving Fuzzy Linear System of Differential Equations
20.2.1 α‐Cut of TFN
20.2.2 Fuzzy Linear System of Differential Equations (FLSDEs)
20.2.3 Solution Procedure for FLSDE
References
Chapter 21 Interval Finite Element Method
21.1 Introduction
21.1.1 Preliminaries
21.1.1.1 Proper and Improper Interval
21.1.1.2 Interval System of Linear Equations
21.1.1.3 Generalized Interval Eigenvalue Problem
21.2 Interval Galerkin FEM
21.3 Structural Analysis Using IFEM
21.3.1 Static Analysis
21.3.2 Dynamic Analysis
References
Index
EULA.

Advanced numerical and semi-analytical methods for differential equations

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